Spectrum of ultrametric Banach algebras of strictly differentiable functions

Abstract

Let IK be an ultrametric complete field and let E be an open subset of IK of strictly positive codiameter. Let D(E) be the Banach IK-algebra of bounded strictly differentiable functions from E to IK, a notion whose definition is detailed. It is shown that all elements of D(E) have a derivative that is continuous in E. Given a positive number r > 0, all functions that are bounded and are analytic in all open disks of diameter r are strictly differentiable. Maximal ideals and continuous multiplicative semi-norms on D(E) are studied by recalling the relation of contiguity on ultrafilters: an equivalence relation. So, the maximal spectrum of D(E) is in bijection with the set of equivalence classes with respect to contiguity. Every prime ideal of D(E) is included in a unique maximal ideal and every prime closed ideal of D(E) is a maximal ideal, hence every continuous multiplicative semi-norm on D(E) has a kernel that is a maximal ideal. If IK is locally compact, every maximal ideal of D(E) is of codimension 1. Every maximal ideal of D(E) is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. The Shilov boundary of D(E) is equal to the whole set of continuous multiplicative semi-norms. Many results are similar to those concerning algebras of uniformly continuous functions but some specific proofs are required.

Ultrafilters and ultrametric Banach algebras of Lipschitz functions

“…First, we generalize results obtained in [8] and [9] to some Banach algebras of uniformly continuous bounded functions which we call semi-compatible and C-compatible and which are related to the contiguity relation yet considered in these papers. Actually, considering such an algebra S, we describe interactions between the contiguity relation defined on ultrafilters on IE and maximal ideals or the multiplicative spectrum of S. We prove that the Shilov boundary of S is the multiplicative spectrum itself.…”

“…x − y has a limit when x and y tend to a separately. Following [9] the functions of E are called strictly differentiable.…”

“…0 ) is a semi-compatible algebra. In [9], it was proved that the algebra here denoted by E is a Banach IK-algebra with respect to the norm . .…”

Finite codimensional maximal ideals in subalgebras of ultrametric uniformly continuous functions

“…In order to prove our main theorems we first need several basic results. In [5], [6], we have proved a proposition that we can easily generalize:…”

“…In [2], [4], [5], [6] we studied several examples of Banach IK-algebras of functions and showed that for each example, each maximal ideal is defined by ultrafilters [1], [7], [8] and that each maximal ideal of finite codimension is of codimension 1: that holds for continuous functions [4] and for all examples of functions we examine in [2], [5], [6]. Thus, we can ask whether this comes from a more general property of Banach IK-algebras of functions, what we will prove here.…”